μ + Knight shift and spin relaxation in indium single crystalKnight shift and spin relaxation in indium single crystal

μ ⁺ SR measurements have been performed in a single crystal indium sample between 12 K and 300 K with a stroboscopic μSR spectrometer. The muonic Knight shiftK _μ and the muonic depolarization rate σ were obtained for various angles θ between the tetragonal crystallinec-axis and the direction of the external field. The isotropic part ofK _μ is only weakly temperature dependent and is consistent with the estimated Pauli spin susceptibility value. At a temperature of 12 K the angular dependence ofM ₂ (the second moment of the field distribution at the muon, obtained from the measured σ(θ) values) allows a clear determination of the muon location — the symmetric tetrahedral site. The observed anisotropicK _μ cannot be explained by the dipoles at the In atoms responsible for the bulk magnetic susceptibility but probably originates from an anisotropic Pauli spin susceptibility.     

Anisotropic μ+ shift in bismuthshift in bismuth

μSR studies in the semi metal Bi have revealed pronounced anisotropies in the μ⁺ Knightshift which vary strongly with temperature. Above 10 K the anisotropy is well described by a P₂(cosθ) distribution with a change of sign around 60 K. Below 10 K we find a very unusual angular dependence in that the anisotropy is described only by terms of fourth order in the direction cosines. Such a dependence is allowed by symmetry arguments, considering that the pointgroup of the crystalline μ⁺ site is D_3d, but requires that the electrons, responsible for the μ⁺ Knight shift, are subject to strong spin orbit coupling or involve effective, field induced moments of orbital origin. The various types of anisotropies seem to be associated with the different site occupations of the μ⁺. The total Knightshift is generally regative which is to be contrasted with the huge positive Knightshift found in the other semimetal Sb /1/.     

Possibility of a Lifshitz phase transition in Cd observed by a singularity in the muon Knight shift

During the past much effort has been devoted to a systematic study of the muon Knight shiftK in metallic environments and its implications on the local electronic structure of hydrogen in metals [1]. These measurements in simple metals were essentially all carried out in polycrystalline samples at room temperature. The present measurements in Cd in polycrystalline and single crystal samples cover a temperature range between 20 K and the melting point of this strongly anisotropic metal (hcp crystal structure,c/a ratio 1.89 — idealc/a ratio 1.63). These measurements add qualitatively new and interesting aspects and insights on the screening of a light hydrogen isotope in a metal as well as on certain properties of the host material itself. The outstanding features of the muon Knight shift in Cd are: (i) a strong intrinsic temperature dependence with an increase ofK of more than 100% between 20 K and the melting point (T=593 K), (ii) an anomaly at 110 K in the form of a singularity in the isotropic part ofK which is interpreted as a band structure effect, (iii) an anisotropic Knight shift contribution fitting the expressionK(T,)=K _iso(T)+K _ax(T) * (3 · cos² –1)/2, where both, the isotropic and the axial contribution ofK , are strongly temperature dependent.     

Muon spin relaxation and knight shift in the heavy-fermion superconductor UPt3

Positive muon spin relaxation experiments have been conducted on the heavy-fermion superconductor UPt₃ in both the normal and superconducting states for zero, transverse, and longitudinally applied magnetic fields. Below 6 K in zero applied field, the μ⁺ relaxation rate is approximately twice that expected from¹⁹⁵Pt nuclear dipolar relaxation alone. Transverse- and longitudinal-field measurements show that the observe relaxation rate depends on magnetic field and is quasistatic in origin. It is suggested that the onset of very weak (≈10⁻³ μ_B/U atom) magnetic ordering below approximately 6 K is responsible for the observed increase in the relaxation rate. μ⁺ Knight shift measurements in the normal state of UPt₃ show a temperature dependent shift K_μ which tracks the bulk susceptibility X. From the K_μ vs. X plot, a μ⁺ hyperfine field of approximately 100 Oe/μ_B is extracted.     

Muon spin relaxation in CeCu2Si2 and muon knight shift in various heavy-fermion systems

Positive muon spin precession has been observed in various heavy-fermion systems in the transverse external magnetic field. In the superconductor CeCu_2.1Si₂, the relaxation rate of muon spins increases rapidly with decreasing temperature below T_C. This is interpreted as the results of the inhomogeneous fields due to the imperfect penetration of the external field into the type-II superconducting state. The magnetic-field penetration depth λ is derived from the observed muon spin relaxation rate. λ is about 1200 ∢ at T∼0.5T_C, and the temperature dependence of λ is consistent with the relation expected for a BCS superconductor. We have also measured the muon Knight shift K_μ in the normal (or paramagnetic) state of various heavy-fermion systems. K_μ is large and negative (about −1000∼−3000 ppm at T=10 K) for CeCu₂Si₂, UPt₃ and CeAl₃, while more complicated signals are measured in CePb₃ and CeB₆. The negative muon Knight shift in the non-magnetic heavy-fermion systems is discussed in terms of the Kondo-coupling between the conduction- and f-electrons.     

NMR and other magnetic resonance techniques in unstable magnets

This paper reviews and compares the use of nuclear magnetic resonance (NMR) and related hyperfine techniques [muon spin rotation (μSR) and, to a lesser extent, other methods] in the study of 4f and 5f magnetism in “unstable magnets”, i.e., intermediate-valent and heavy-fermion materials. In both NMR and μSR the features of interest are the spectral shape, the frequency shiftK (Knight shift in metals) and the spin-lattice relaxation rate 1/T ₁. For temperatures below the characteristic or “Kondo” temperatureT ₀ these experiments given evidence for (1) modification of the transferred hyperfine field [nonlinearK(χ)]. (2) spin fluctuations with a characteristic fluctuation rate ∼k _B T ₀/h, (3) strong energy-gap anisotropy (zeros of the gap along lines on the Fermi surface) in heavy-fermion superconductors, (4) spin-singlet Cooper pairing from the change in muon Knight shift in superconducting UBe₁₃, and (5) very weak static magnetism (10⁻¹–10⁻³ μ_B/f atom) in CeAl₃, CeCu₂Si₂, U_1−x Th _x Be₁₃ (x=0.033), and UPt₃. There is some controversy concerning the interpretation of 1/T ₁ well aboveT ₀ in UBe₁₃; the situation is reviewed.     

μ+‐site in heavy‐fermion compounds

We report on the μ⁺‐site determination in different heavy‐fermion systems, resulting from the analysis of the angular dependence of parameters of the μSR spectra. After an introduction on the basics of the analysis of the μ⁺‐Knight shift, we briefly discuss the specific information accessible once the knowledge of the crystallographic site occupied by the implanted muon is gained. This revised version was published online in July 2006 with corrections to the Cover Date.     

A comparative study of the magnetic heavy-electron materials U2Zn17 and UCu5 by μ+SR

A systematic μ⁺SR study of the magnetic heavy-electron systems U₂Zn₁₇ and UCu₅ in the paramagnetic and in the magnetically ordered state is presented. In both systems the antiferromagnetic nature of the low-temperature phase could be at least partially confirmed, but the muon reveals significant differences with regard to the phase transition itself. UCu₅ behaves like a model-antiferromagnet showing a drastic increase of the relaxation rate both below and above T_N, two spontaneous frequencies in the ordered phase, and a Knight shift above T_N which scales with the bulk susceptibility. In contrast U₂Zn₁₇ shows a loss of μ⁺ asymmetry by 20% below T_N, which is independent of the external field but can be quenched in sufficiently strong longitudinal fields. No scaling of the Knight shift and the susceptibility was observed and no critical increase of the relaxation rate λ. Most astonishing is the strong and nonlinear field dependence of λ above and below T_N in both compounds. The absence of longitudinal relaxation demonstrates the static origin of λ.     

On the Boltzmann Equation for Fermi–Dirac Particles with Very Soft Potentials: Averaging Compactness of Weak Solutions

The paper considers macroscopic behavior of a Fermi–Dirac particle system. We prove the L ¹-compactness of velocity averages of weak solutions of the Boltzmann equation for Fermi–Dirac particles in a periodic box with the collision kernel b(cos θ)|ρ−ρ _*|^γ, which corresponds to very soft potentials: −5 < γ ≤ −3 with a weak angular cutoff: ∫₀ ^π b(cos θ)sin ³θ dθ < ∞. Our proof for the averaging compactness is based on the entropy inequality, Hausdorff–Young inequality, the L ^∞-bounds of the solutions, and a specific property of the value-range of the exponent γ. Once such an averaging compactness is proven, the proof of the existence of weak solutions will be relatively easy.     

B→K1?+?? decays in a family non-universal Z′ model

The implications of the family non-universal Z′ model in the B→K ₁(1270,1400)ℓ ⁺ ℓ ⁻(ℓ=e ,μ ,τ) decays are explored, where the mass eigenstates K ₁(1270, 1400) are the mixtures of ¹ P ₁ and ³ P ₁ states with the mixing angle θ. In this work, considering the Z′ boson and setting the mixing angle θ=(−34±13)^○, we analyze the branching ratio, the dilepton invariant mass spectrum, the normalized forward–backward asymmetry and lepton polarization asymmetries of each decay mode. We find that all observables of B→K ₁(1270)μ ⁺ μ ⁻ are sensitive to the Z′ contribution. Moreover, the observables of B→K ₁(1400)μ ⁺ μ ⁻ have a relatively strong θ-dependence; thus, the Z′ contribution will be buried by the uncertainty of the mixing angle θ. Furthermore, the zero crossing position in the FBA spectrum of B→K ₁(1270)μ ⁺ μ ⁻ at low dilepton mass will move to the positive direction with Z′ contribution. For the tau modes, the effects of Z′ are not remarkable due to the small phase space. These results could be tested in the running LHC-b experiment and Super-B factory.     

μ+ shift study inCdMgx (x=1%, 2%, 2.5%)shift study inCdMgx (x=1%, 2%, 2.5%)

In continuation of our earlier studies of the anomalous temperature dependence of the μ⁺ Knight shift K_μ in Cd,CdHg (1.2 at %) andCdMg (3.38 at %) — interpreted as due to Van Hove-type singularities in the local density of electron states /1/ — we have studied the temperature dependence of the μ⁺ Knight shift in polycrystallineCdMg (1.04 at %, 2.05 at %) and in a single crystal ofCdMg (2.5 at %). In contrast to pure Cd no anisotropies in K_μ could be detected. The temperature dependence of K_μ in aCdMg (2.05 at %) sample and in the monocrystallineCdMg (2.5 at %) sample essentially reproduces the one previously observed in polycrystallineCdMg (3.38 at %), showing a steplike discontinuity and a logarithmic singularity. A complete different behaviour is observed inCdMg (1.04 at %), where no logarithmic singularity seems to show up and where a steplike discontinuity of opposite sign at around 90 K is clearly seen. If these singularities are still to be interpreted in terms of Van Hove singularities, the question arises why there is such a nonlinear dependence on the Mg concentration.     

Electric field influence on spin polarization vector behaviour of the negative muon in silicon

When a negatively charged muon is captured by a silicon atom, the atom is transformed into a solitary acceptor center similar to an Al atom. An external electric field influences the formation process of the neutral acceptor center (A.C.). It is shown in this article that the behaviour of the muon polarization vector changes appreciably in electric fields with intensities E\,\gtrsim\,10 V/cm. We estimate the muon spin relaxation rate \varLambda_μ in the so‐called “dirty limit”; for example, interaction between an internal electric field from charged impurities and a nonzero dipole moment of the A.C. is taken into account. A phonon mechanism is proposed to explain the temperature dependence of \varLambda_μ. We also estimate the value of the paramagnetic shift of the muon spin precession frequency \delta\omega/\omega₀ which is also temperature dependent. This revised version was published online in August 2006 with corrections to the Cover Date.     

Magnetic studies with μSR at high pressures

A newly developed high pressure-low temperature μSR spectrometer was employed for two types of magnetic studies. Firstly we measured the pressure dependence of the local magnetic field B_μ in Fe, Co and Ni at 77 K up to 7 kbar. From the pressure derivative dlnB_μ/dP the volume derivative dlnB _Hf /dlnV was deduced. In connection with previous room temperature data we calculated the temperature dependence of B_hf at constant volume. It deviates markedly from the temperature dependence of the host magnetization. Secondly, we looked at the pressure dependence of the muonic Knight shift in Sb at 30 K for polycrystalline and single crystal samples. A strong pressure dependence was observed which depends on the orientation of the magnetic field relative to the c-axis. The pressure coefficients of the isotropic and the axial term of the Knight shift were deduced.     

μ–H Interaction in Hydrogen Bonding Systems

Muon spin rotation (μSR) experiments were performed on single crystal samples of KH₂PO₄(KDP) and KD₂PO₄(dKDP) to study the dynamics of hydrogen in hydrogen bonding systems. At low temperature, the nuclear dipole interaction of muon and proton was confirmed from the angular dependence of precession frequency of the muon spin under zero magnetic field. The muon occupation site was also determined. A clear change in μSR spectra was observed at the antiferroelectric transition temperature (123 K). At 90 K well below the transition temperature, the muon spin starts to relax, possibly due to muon dynamics. This revised version was published online in September 2006 with corrections to the Cover Date.     

Muon knight shift in platinum and in β-PdHx (x=0.70, 0.75, 0.86)

The ⁺ Knight shift in Platinum has been measured between 20 K and 785 K. It shows a strong temperature dependence and scales with the magnetic bulk susceptibility. A temperature independent contribution of +53±15 ppm and a d-electron induced hyperfine field per unpaired d-electron per atom of B _hfd ^a =–5.03 kG(±8.5%) are obtained. The ⁺ Knight shift in PdH_0.70, PdH_0.75 and PdH_0.86 shows no dependence on temperature between 20 K and RT and increases from K=–(8±3) ppm for x=0.70 to K =+(6.5±3) ppm K=+(6.5±3) ppm for x=0.86, in good agreement with proton Knight shift measurements.     

μ+knight shifts and trapping in diluteSbSn alloys

Giant μ⁺ Knight shifts K_μ have been studied previously in Sb andSbBi alloys. Here we report μ⁺SR measurements on Sb with dilute heterovalent Sn impurities. A dramatic concentration dependence is observed: K_μ is increased slightly (relative to the value in pure Sb) by Sn concentrations of <0.1%, whereas larger concentrations cause a drastic reduction of K_μ. One concern could be that K_μ values in the alloy might reflect local band structure in trap sites near Sn impurities rather than the bulk “host” band structure of the alloy. This is indeed the case in bothSbSn (0.03%) andSbSn (0.06%), where a second, lower frequency TF-μSR signal begins to appear for T>60K. The amplitude of the low K_μ signal grows with increasing T at the expense of the amplitude of the high-K_μ, low-T signal, suggesting that the μ⁺ migrates through the host lattice to trap sites. A simple trapping model correctly describes the observed T-dependence of the amplitudes, phases and relaxation rates of the two signals. We conclude that the low-T Knight shift is truly characteristic of the host band structure while the much lower K_μ value of the high-T site is characteristic of a specific trap site, presumably adjacent to a Sn impurity.     

Muon Knight shift in platinum and in β-PdHx (x=0.70, 0.75)

The ⁺ Knight shift in platinum has been measured between 20 K and 785 K. It shows a strong temperature dependence and scales with the magnetic bulk susceptibility. A temperature independent contribution of +40 to +60 ppm and a d-electron induced hyperfine field per unpaired d-electron per atom ofB _hf,d =–5.03 kG (±8.5%) are obtained. The ⁺ Knight shift in PdH_0.70 and PdH_0.75 shows no dependence on temperature between 20 K andRT and increases fromK ppm forx=0.70 toK ppm forx=0.75, in good agreement with proton Knight shift measurements.     

Pressure dependence of the muon knight shift in antimony

The pressure dependence of the muon Knight shift in antimony has been measured at 30K using polycrystalline samples and at 10K using single crystals. A considerably stronger pressure dependence is observed with the field parallel to the c-axis than perpendicular. The deduced linear parts of the isotropic and axial pressure dependence are dK_iso/dP=−0.19(3)%/kbar and dK_ax/dP−0.24 (5)%/kbar. First principle molecular-cluster calculations show the origin, of the huge Knight shift and its pressure dependence.     

Diamagnetism and muon knight shift in semimetallic and semiconducting BiSb single crystals

A series of single crystalline Bi_1−x Sb _x alloys (x=0,0.03, 0.14, 0.19, 0.37) covering both the semimetallic (0≤x≤0.07 orx≥0.22) and the semiconducting region (0.07≤x≤0.22) was examined using the stroboscopic μ⁺ SR method. The μ⁺ Knight shift, negative for all Sb concentrations, shows pronounced temperature dependences and large anisotropies. A scaling with the-negative-total magnetic susceptibility [1] is found in the semiconducting alloys. In detail, the isotropic part of the μ⁺ Knight shift is proportional to the isotropic part of the susceptibility, and the anisotropic Knight shift scales with the anisotropic susceptibility. Possible mechanisms leading to this relation are investigated.     

Phenomenology of the minimal<Emphasis Type="Italic">SO</Emphasis>(10) SUSY model

In this talk I define what I call the minimalSO(10) SUSY model. I then discuss the phenomenological consequences of this theory, vis-a-vis gauge and Yukawa coupling unification, Higgs and super-particle masses, the anomalous magnetic moment of the muon, the decayB _s → μ⁺ itμ⁻ and dark matter. On leave of absence at the Institute for Advanced Study, Princeton, NJ, USA